The movement of a material point in space is given by the equations x = 2 + 4t ^ 2 y = 3t

The movement of a material point in space is given by the equations x = 2 + 4t ^ 2 y = 3t z = 3t + 4t ^ 2. Find the modules of the radius vector of velocity and acceleration at time t = 2 s

The movement of a material point in space is given by the equations x = 2 + 4t ^ 2 y = 3t z = 3t + 4t ^ 2. Find the modules of the radius vector of the velocity and acceleration at the moment of time t = 2 s. We will assume that the coordinates are given in meters.
Then the modulus is the radius of the vector:
Coordinates at the moment t = 2 s:
x = 2 + 4t² = 2 + 16 = 18 m.
y = 3t = 6 m.
z = 3t + 4t² = 6 + 16 = 22 m.
Radius vector modulus r:
r = root (x² + y² + z²) = root (18² + 6² + 22²) = 29.05 m.
We find the components of the velocity vector as time derivatives of the coordinates of the point:
Vx = (2 + 4t²) ´ = 8t = 16 m / s;
Vy = (3t) ´ = 3 m / s;
Vz = (3t + 4t²) ´ = 3 + 8t = 19 m / s
Velocity module V:
V = √ (Vx ^ 2 + Vx ^ 2 + Vx ^ 2) = √ (16² + 3² + 19²) = 25 m / s.
We find the components of the acceleration vector as derivatives of the components of the velocity:
Ax = (8t) ´ = 8 m / s²;
Ay = (3) ´ = 0 m / s²;
Az = (3 + 8t) ´ = 8 m / s²;
Acceleration module:
A = √ (Ax ^ 2 + Ay ^ 2 + Az ^ ​​2) = √ (8² + 0 + 8²) = 11.31.

Answer: The modulus of the radius vector is 29 m, the modulus of the velocity is 25 m / s, and the modulus of acceleration is 11 m / s².